Pattern Avoidance for Random Permutations

TL;DR

This paper applies Poisson approximation techniques to derive explicit error bounds for pattern avoidance in random permutations, including Mallows permutations, enabling precise probability estimates for pattern occurrences.

Contribution

It introduces explicit error bounds for pattern avoidance probabilities in random permutations using Poisson approximation, extending to Mallows permutations.

Findings

Derived explicit error bounds for pattern avoidance probabilities.

Established Poisson approximation for pattern occurrence distributions.

Analyzed pattern occurrences in Mallows permutations.

Abstract

Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences and a corresponding joint distribution of independent Bernoulli random variables, which as a corollary yields a Poisson approximation for the distribution of the number of occurrences of any pattern. We also investigate occurrences of consecutive patterns in random Mallows permutations, of which uniform random permutations are a special case. These bounds allow us to estimate the probability that a pattern occurs any number of times and, in particular, the probability that a random permutation avoids a given pattern.